Multi-layer Global Routing Considering Via and Wire Capacities

1. Multi-layer Global Routing Considering Via and Wire Capacities Chin-Hsiung Hsu, Huang-Yu Chen Yao-Wen ChangICCAD.2008

2. Outline • Introduction • Problem Formulation • Routing Methodology • Experimental Results • Conclusion

3. Introduction • TheISPD2007 contest released two sets of benchmarks for 2D and 3D globalrouting and defined a performance cost metricbased on the prioritized order : • The total overflow • Themaximum overflow • The total wirelength • The 2008 contestfurther considered the trade-off between wirelength and runtime.

4. Introduction • But thecontest without considering some crucial design rulessuch as vias for routing through multiple metal layers. • Which mislead the participants to directly map their 2D routing solutions to 3D ones by layer assignment. • These simplifications may not be realistic for practical applications.

5. Introduction • All these routers ignore the residual via capacity in a routed region, which might complicate subsequent detailed routing.

6. Outline • Introduction • Problem Formulation • Routing Methodology • Experimental Results • Conclusion

7. Problem Formulation • For global routing, the routing region is partitioned into tiles and edges models the routing region. • The global tile node represents a tile, and the global edge models the relationship between adjacent tiles. Tiles => global tile nodes Edges => global edges

8. Problem Formulation • To address the via-capacity issue, the via capacity of a tile t as • By this model, the via resource of a global routing tile t depends on the usage of wires but does not affect the wire capacity of t. vw :the via width vs :via spacing aw :the probabilistic area of wires passing through the tile boundary at :the area of the tile t ao :the total area of the obstacles in t ai :the area of in-tile nets in t

9. Problem Formulation • The 3D Routing Problem: Given a netlist, width and spacing of vias and wires, and wire capacities, find 3D routing paths connect-ing pins of each net such that the total number of wire and via overflow, and the total wirelength are minimized.

10. Outline • Introduction • Problem Formulation • Routing Methodology • Experimental Results • Conclusion

11. Algorithm Preview

12. Algorithm Preview • Initially, each net is decomposed into 2-pin connections by the minimum spanning tree (MST) to maximize the flexibility of the topology • Then, apply least-flexibility-first routing to route the nets with lower flexibility to guide the subsequent monotonic routing in a look-ahead manner. Routing Result

13. Algorithm Preview • If the current 2D routing does not incur any 2D (wire) overflow, layer assignment is performed to produce an initial 3D routing solution. • If the initial 3D routing solution also does not cause any 3D (wire and via) overflow, the final global-routing solution is obtained. Routing Result

14. Algorithm Preview • If the routing solution is not overflow-free, an iterative negotiation-based rip-up and rerouting scheme is performed to find a minimal-overflow solution for 2D and 3D routing. Routing Result

15. Routing Methodology • Least-Flexibility-First Routing • Negotiation-Based Rip-Up and Rerouting

16. A. Least-Flexibility-First Routing • It first route the nets located in a congested region with higher pin density or with shorter wirelength since they enjoy less routing flexibility than longer ones. • This enables the subsequent routing to quickly predict the congestion hot spots for overflow reduction.

17. B. Negotiation-Based Rip-Up and Rerouting • Multi-Source Multi-Sink Aerial-Monotonic Routing (MSAMR) • The (3D) Aerial-Monotonic Routing (AMR) • Multi-Source Multi-Sink Routing

18. Routing Methodology • In 3D routing graph, the z-directionglobal edge connecting two global tiles on two layers can be upward and downward, and thus cycles may be introduced. • As a result, existing linear-time algorithms that find the shortest path on a directed acyclic graph is not applicable to this case.

19. The AMR algorithm iteratively performs the two phase operations of the pile update and pile propagation • A pile represents a set of global tile nodes in the z-direction of a global routing graph • AMR algorithm finds an optimal aerial-monotonic the routing path in O(V) time V : the number of nodes

20. Routing Methodology • With the increasing number of congestedregions, the multi-source and multi-sink routing can effectively find alternative paths to connect two subtrees. • The multi-source multi-sink maze routing [15] requires O(V·lgV) time and becomes very slow when V is large, because it spends significant time in maintaining a large priority queue. V : the number of nodes

21. Routing Methodology • The MSAMR algorithm first classify the topological relationship of the two subtrees into four categories: 1. xy non-overlap 2. x overlap 3. y overlap 4. xy overlap

22. Routing Methodology • Then, AMR selects a minimum-cost node from the nodes on the target tree Ttin the routing box with a specific direction, and then back traces to any node in the source sub-tree Ts. • The MSAMR algorithm can find an optimal 3D multi-source multi-sink aerial-monotonic routing path in only O(V) time.

23. B. Negotiation-Based Rip-Up and Rerouting • Multi-Source Multi-Sink Escaping-Point Routing • Two-phase Aerial-Monotonic Routing (TAMR) • Two-phase Multi-Source Multi-Sink Aerial-Monotonic Routing (TMSAMR)

24. Routing Methodology • To find an optimal escaping-point routing path, the traditional bidirectional search stops when the two search paths meet in the middle, and then it finds a path just like the path found by the unidirectional search. • In contrast, two-phase aerial-monotonic routing (TAMR) can find a routing path allowing aerial detours while the unidirectional search cannot.

25. Routing Methodology • The two-phase aerial-monotonic routing (TAMR) algorithm first decide the size of box according to the user-specified number ext. • And four 3D aerial-monotonic propagations are performed from the source and target to the four corner points of the extended routing box.

26. Routing Methodology • Finally, find a node with the minimum distanceand back trace to the source and the target from the node with the minimum distance.

27. Routing Methodology • We can extend the TAMR algorithm to the TMSAMR one by performing the two-phase MSAMR algorithm from both the source and the target subtrees according to their topological relation.

28. Outline • Introduction • Problem Formulation • Routing Methodology • Experimental Results • Conclusion

29. Experimental Results • In the ISPD’07 global routing benchmarks, turn off the via-capacity and in-tile-net considerations to make fair comparisons. • OF : the number of overflow • WL : the total wirelength • Global Routing Results without Considering Via Capacity

30. Experimental Results • This result shows the comparisons of via-aware 3D global routing results on the ISPD’07 benchmarks. • VOF : reports the number of via overflow • Via-Aware Global Routing Results

31. Conclusion • This paper have derived a congestion metric, dynamic via capacity, for global routing. • Two routing algorithms, least-flexibility-first routing and multi-source multi-sink escaping-point routing,for congestion optimization. • The linear-time escaping-point routing algorithm is optimal.